It Is NL-complete to Decide Whether a Hairpin Completion of Regular Languages Is Regular

The hairpin completion is an operation on formal languages which is inspired by the hairpin formation in biochemistry. Hairpin formations occur naturally within DNA-computing. It has been known that the hairpin completion of a regular language is linear context-free, but not regular, in general. However, for some time it is was open whether the regularity of the hairpin completion of a regular language is is decidable. In 2009 this decidability problem has been solved positively by providing a polynomial time algorithm. In this paper we improve the complexity bound by showing that the decision problem is actually NL-complete. This complexity bound holds for both, the one-sided and the two-sided hairpin completions

Decision Problems for Deterministic Pushdown Automata on Infinite Words

The article surveys some decidability results for DPDAs on infinite words (omega-DPDA). We summarize some recent results on the decidability of the regularity and the equivalence problem for the class of weak omega-DPDAs. Furthermore, we present some new results on the parity index problem for omega-DPDAs. For the specification of a parity condition, the states of the omega-DPDA are assigned priorities (natural numbers), and a run is accepting if the highest priority that appears infinitely often during a run is even. The basic simplification question asks whether one can determine the minimal number of priorities that are needed to accept the language of a given omega-DPDA. We provide some decidability results on variations of this question for some classes of omega-DPDAs.Comment: In Proceedings AFL 2014, arXiv:1405.527

Reachability in Higher-Order-Counters

Higher-order counter automata (\HOCS) can be either seen as a restriction of higher-order pushdown automata (\HOPS) to a unary stack alphabet, or as an extension of counter automata to higher levels. We distinguish two principal kinds of \HOCS: those that can test whether the topmost counter value is zero and those which cannot. We show that control-state reachability for level $k$ \HOCS with $0$-test is complete for \mbox{$(k-2)$}-fold exponential space; leaving out the $0$-test leads to completeness for \mbox{$(k-2)$}-fold exponential time. Restricting \HOCS (without $0$-test) to level $2$, we prove that global (forward or backward) reachability analysis is \PTIME-complete. This enhances the known result for pushdown systems which are subsumed by level $2$ \HOCS without $0$-test. We transfer our results to the formal language setting. Assuming that \PTIME \subsetneq \PSPACE \subsetneq \mathbf{EXPTIME}, we apply proof ideas of Engelfriet and conclude that the hierarchies of languages of \HOPS and of \HOCS form strictly interleaving hierarchies. Interestingly, Engelfriet's constructions also allow to conclude immediately that the hierarchy of collapsible pushdown languages is strict level-by-level due to the existing complexity results for reachability on collapsible pushdown graphs. This answers an open question independently asked by Parys and by Kobayashi.Comment: Version with Full Proofs of a paper that appears at MFCS 201

Using formal methods to support testing

Formal methods and testing are two important approaches that assist in the development of high quality software. While traditionally these approaches have been seen as rivals, in recent years a new consensus has developed in which they are seen as complementary. This article reviews the state of the art regarding ways in which the presence of a formal specification can be used to assist testing

Mutation of Directed Graphs -- Corresponding Regular Expressions and Complexity of Their Generation

Directed graphs (DG), interpreted as state transition diagrams, are traditionally used to represent finite-state automata (FSA). In the context of formal languages, both FSA and regular expressions (RE) are equivalent in that they accept and generate, respectively, type-3 (regular) languages. Based on our previous work, this paper analyzes effects of graph manipulations on corresponding RE. In this present, starting stage we assume that the DG under consideration contains no cycles. Graph manipulation is performed by deleting or inserting of nodes or arcs. Combined and/or multiple application of these basic operators enable a great variety of transformations of DG (and corresponding RE) that can be seen as mutants of the original DG (and corresponding RE). DG are popular for modeling complex systems; however they easily become intractable if the system under consideration is complex and/or large. In such situations, we propose to switch to corresponding RE in order to benefit from their compact format for modeling and algebraic operations for analysis. The results of the study are of great potential interest to mutation testing

On Pebble Automata for Data Languages with Decidable Emptiness Problem

In this paper we study a subclass of pebble automata (PA) for data languages for which the emptiness problem is decidable. Namely, we introduce the so-called top view weak PA. Roughly speaking, top view weak PA are weak PA where the equality test is performed only between the data values seen by the two most recently placed pebbles. The emptiness problem for this model is decidable. We also show that it is robust: alternating, nondeterministic and deterministic top view weak PA have the same recognition power. Moreover, this model is strong enough to accept all data languages expressible in Linear Temporal Logic with the future-time operators, augmented with one register freeze quantifier.Comment: An extended abstract of this work has been published in the proceedings of the 34th International Symposium on Mathematical Foundations of Computer Science (MFCS) 2009}, Springer, Lecture Notes in Computer Science 5734, pages 712-72

Decision Problems for Subclasses of Rational Relations over Finite and Infinite Words

We consider decision problems for relations over finite and infinite words defined by finite automata. We prove that the equivalence problem for binary deterministic rational relations over infinite words is undecidable in contrast to the case of finite words, where the problem is decidable. Furthermore, we show that it is decidable in doubly exponential time for an automatic relation over infinite words whether it is a recognizable relation. We also revisit this problem in the context of finite words and improve the complexity of the decision procedure to single exponential time. The procedure is based on a polynomial time regularity test for deterministic visibly pushdown automata, which is a result of independent interest.Comment: v1: 31 pages, submitted to DMTCS, extended version of the paper with the same title published in the conference proceedings of FCT 2017; v2: 32 pages, minor revision of v1 (DMTCS review process), results unchanged; v3: 32 pages, enabled hyperref for Figure 1; v4: 32 pages, add reference for known complexity results for the slenderness problem; v5: 32 pages, added DMTCS metadat